Research is one of those things that is a boon to an application, but it's not an absolute requirement *unless* you are applying to a research-oriented program like an MD/PhD or MD/MS program. With those exceptions, even the more research-oriented schools still have a significant minority of matriculants who have not participated in research. That being said, it's always good to know your audience. Most adcom members are MD/DO clinicians, MD/DO researchers, or PhD researchers. So there may indeed be a strong bias toward candidates with research experience simply because of the makeup of the adcom. Keep in mind too that there is a bias toward applicants who have participated in research because the best candidates for med school tend to come from institutions where research experience is available to undergrads, and many of these students do choose to participate when the opportunity is offered to them. Successful candidates are successful because they are hard workers in general, including making the effort to get involved in ECs like research that they perceive as being valued by adcoms. So research as an EC is also a marker of motivation/work ethic.
As someone with extensive graduate training in research (both clinical and basic), I'm going to respectfully disagree with your adcom friend here. Participating in research at the UG level does not teach these skills to participants. It would be great if it did, but I sat through plenty of journal clubs at my highly respected academic residency where, outside of a few of us who were interested in EBM, no one else had any experience (or even much interest) in critically evaluating the primary studies. That's why there is so much secondary literature where someone else who knows what they're doing has already performed this evaluation work. (Think of commonly used clinical resources such as the Cochrane Collaboration reviews, UpToDate, specialty expert guidelines, etc.) What UG research participation does do, however, is two other important things: 1) give participants an understanding for how knowledge in the sciences is actually obtained (slowly, painfully, after many wrong turns), and 2) in some cases, lead someone who hadn't considered an academic career to become an academic. I got interested in research after participating in UG research experience; I certainly did not start college intending to go to graduate school. This is probably fairly common.
Yes. Other networking ideas include the career center or premed club at your university or local med school. If you're willing to make enough of a time commitment where it's worth the effort of training you, someone will likely be willing to take you on, especially if you're looking to volunteer in a lab. But again, unless you are applying for combined degree programs with a mission to train physician scientists, you don't absolutely have to have research experience. It's not true that everyone does.